Stability and phase error analysis of FD-TD in dispersive dielectrics

Four FD-TD extensions for the modeling of pulse propagation in Debye or Lorentz dispersive media are analyzed through studying the stability and phase error properties of the coupled difference equations corresponding to Maxwell´s equations and to the equations for the dispersion. For good overall accuracy the author shows that all schemes should be run at their Courant stability limit, and that the timestep should finely resolve the medium timescales. Particularly, for Debye schemes it should be at least Δt=10^-3τ, while for Lorentz schemes it should be Δt=10^-2τ, where τ is a typical medium relaxation time. A numerical experiment with a Debye medium confirms this. The author has determined that two of the discretizations for Debye media are totally equivalent. In the Lorentz medium case the author establishes that the method that uses the polarization differential equation to model dispersion is stable for all wavenumbers, and that the method using the local-in-time constitutive relation is weakly unstable for modes with wavenumber k such that kΔx>π/2